Linear search problem
Encyclopedia
In computational complexity theory
, the Linear search problem is an optimal search problem introduced by Richard E. Bellman. (independently considered by Anatole Beck ).
The linear search problem for a general probability distribution is unsolved yet. However, there exists a dynamic programming
algorithm that produces a solution for any discrete distribution and also an approximate solution, for any probability distribution, with any desired accuracy.
The linear search problem was solved by Anatole Beck and Donald J. Newman
(1970) as a two-person zero-sum game. Their minimax trajectory is to double the distance on each step and the optimal strategy is a mixture of trajectories that increase the distance by some fixed constant . This solution gives search strategies that are not sensitive to assumptions concerning the distribution of the target. Thus, it also presents an upper bound for a worst case scenario. This solution was obtained in the framework of an online algorithm
by Shmuel Gal
. . The best online competitive ratio is 9 but it can be reduced to 4.6 by using a randomized strategy. The online solution with a turn cost is given by.
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
, the Linear search problem is an optimal search problem introduced by Richard E. Bellman. (independently considered by Anatole Beck ).
The problem
"An immobile hider is located on the real line according to a known probability distribution. A searcher, whose maximal velocity is one, starts from the origin and wishes to discover the hider in minimal expected time. It is assumed that the searcher can change the direction of his motion without any loss of time. It is also assumed that the searcher cannot see the hider until he actually reaches the point at which the hider is located and the time elapsed until this moment is the duration of the game." It is obvious that in order to find the hider the searcher has to go a distance x1 in one direction, return to the origin and go distance x2 in the other direction etc., (the length of the n-th step being denoted by xn), and to do it in an optimal way. (However, an optimal solution need not have a first step and could start with an infinite number of small 'oscillations'.) This problem is usually called the linear search problem and a search plan is called a trajectory. It has attracted much research, some of it quite recent. (Especially Beck, e.g.,.)The linear search problem for a general probability distribution is unsolved yet. However, there exists a dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
algorithm that produces a solution for any discrete distribution and also an approximate solution, for any probability distribution, with any desired accuracy.
The linear search problem was solved by Anatole Beck and Donald J. Newman
Donald J. Newman
Donald J. Newman was an American mathematician and professor, excelling at the Putnam mathematics competition while an undergraduate at City College of New York and New York University, and later receiving his PhD from Harvard University in 1953.- Life and works :Newman was born in Brooklyn, New...
(1970) as a two-person zero-sum game. Their minimax trajectory is to double the distance on each step and the optimal strategy is a mixture of trajectories that increase the distance by some fixed constant . This solution gives search strategies that are not sensitive to assumptions concerning the distribution of the target. Thus, it also presents an upper bound for a worst case scenario. This solution was obtained in the framework of an online algorithm
Online algorithm
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from...
by Shmuel Gal
Shmuel Gal
Shmuel Gal is a professor of statistics at the University of Haifa in Israel.Gal received a Ph.D. in mathematics from the Hebrew University of Jerusalem....
. . The best online competitive ratio is 9 but it can be reduced to 4.6 by using a randomized strategy. The online solution with a turn cost is given by.